Subgroups — #14541860

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763533	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763528	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763202	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763190	C₂	1	1	yes	maximal, characteristic
5	#11763033	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763029	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763026	C₂	1	1	yes	maximal, characteristic
8	#11763021	C₂	1	1	yes	maximal, characteristic
9	#11763017	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763016	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762911	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762910	C₂	1	1	yes	maximal, characteristic, complemented
14	Q₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#497	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₇×#124	C₂²	1	1	yes	characteristic
22	#10587633	C₂²	1	1	yes	characteristic
23	#10587633	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
27	#10587630	C₂²	1	1	yes	characteristic
28	C₂×#63231	C₂²	1	1	yes	characteristic
29	C₂×#63231	C₂²	1	1	yes	characteristic
30	C₂×#63230	C₂²	1	1	yes	characteristic
31	#10587622	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
33	C₂×#63228	C₂²	1	1	yes	characteristic
34	C₂×#63219	C₂²	1	1	yes	characteristic
35	#10587584	—	2	1	no	—
36	#10587584	—	2	1	no	—
37	#10587583	—	2	1	no	—
38	#10587578	—	2	1	no	—
39	#10587565	—	2	1	no	—
40	#10587565	—	2	1	no	—
41	#10587564	—	2	1	no	—
42	#10587563	—	2	1	no	—
43	C₄×D₆₈	C₂²	1	1	yes	characteristic
44	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
45	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
46	C₁₇×#97	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587506	C₂²	1	1	yes	characteristic
49	#10587506	C₂²	1	1	yes	characteristic
50	#10587503	C₂²	1	1	yes	characteristic
51	#10587492	C₂²	1	1	yes	characteristic
52	#10587489	C₂²	1	1	yes	characteristic
53	#10587487	C₂²	1	1	yes	characteristic
54	#10587486	C₂²	1	1	yes	characteristic
55	#10587485	C₂²	1	1	yes	characteristic
56	#10587485	C₂²	1	1	yes	characteristic
57	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₇	—	2	1	no	—
64	C₂×D₆₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×Dic₃₄	D₄	1	1	yes	characteristic
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
68	Q₁₆×C₁₇	—	2	2	no	—
69	Q₁₆×C₁₇	—	4	1	no	—
70	SD₁₆×C₁₇	—	2	2	no	—
71	SD₁₆×C₁₇	—	4	1	no	—
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	—	2	1	no	—
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₂²⋊C₄×C₁₇	—	2	1	no	—
77	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₄²×C₁₇	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₁₇	—	2	1	no	—
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	#63231	—	4	1	no	—
83	#63231	—	4	1	no	—
84	#63230	—	4	1	no	—
85	#63228	—	4	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	C₄⋊Dic₁₇	—	2	1	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
93	C₄⋊Dic₁₇	—	2	1	no	—
94	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	D₄	1	1	yes	characteristic
101	#63225	D₄	1	1	yes	characteristic
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₄×Dic₁₇	—	2	1	no	—
105	C₄×Dic₁₇	—	2	1	no	—
106	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
107	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	C₂×#3622	C₂³	1	1	yes	characteristic
110	#63219	—	2	2	no	—
111	C₂³×C₁₇	—	2	1	no	—
112	C₂²×D₁₇	—	2	1	no	—
113	Q₈×C₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	2	1	no	—
119	D₄×C₁₇	—	2	1	no	—
120	D₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	4	2	no	—
131	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
135	D₆₈	—	2	1	no	—
136	D₆₈	—	2	1	no	—
137	C₄×D₁₇	—	4	2	no	—
138	Dic₃₄	—	2	1	no	—
139	Dic₃₄	—	2	1	no	—
140	Dic₃₄	—	4	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	#41	1	2	yes	—
143	#3622	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3402	—	17	1	no	maximal, Sylow
146	C₂²×C₁₇	—	4	1	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	—	4	1	no	—
149	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₇	—	4	1	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₄×C₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
160	Dic₁₇	—	2	1	no	—
161	Dic₁₇	—	2	1	no	—
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	#556	—	17	1	no	—
168	#556	—	17	1	no	—
169	#497	—	17	1	no	—
170	#497	D₁₇	1	1	yes	characteristic, complemented
171	#495	—	17	1	no	—
172	#464	—	17	1	no	—
173	#464	—	17	1	no	—
174	#463	—	17	1	no	—
175	#463	—	17	1	no	—
176	#462	—	17	1	no	—
177	#462	—	17	1	no	—
178	#446	—	17	1	no	—
179	#440	—	17	1	no	—
180	#440	—	17	1	no	—
181	C₄×Q₁₆	—	17	1	no	—
182	C₂×C₁₇	—	8	1	no	—
183	C₂×C₁₇	#550	1	1	yes	characteristic
184	C₂×C₁₇	#575	1	1	yes	characteristic
185	C₂×C₁₇	#578	1	1	yes	characteristic
186	D₁₇	—	8	1	no	—
187	C₂×Q₁₆	—	17	1	no	—
188	C₂×Q₁₆	—	17	1	no	—
189	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
190	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
191	C₂×SD₁₆	—	17	1	no	—
192	C₂×SD₁₆	—	17	1	no	—
193	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
194	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
195	C₂×D₈	—	17	1	no	—
196	C₄⋊Q₈	—	17	1	no	—
197	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
198	#126	—	34	1	no	—
199	#126	—	34	1	no	—
200	#126	—	34	1	no	—
201	#126	—	34	1	no	—
202	#124	—	17	1	no	—
203	#124	—	17	1	no	—
204	#124	—	17	1	no	—
205	#124	C₂×D₁₇	1	1	yes	characteristic
206	C₂²⋊Q₈	—	34	1	no	—
207	C₂²⋊Q₈	—	34	1	no	—
208	C₄×Q₈	—	17	1	no	—
209	C₄×Q₈	—	17	1	no	—
210	C₄×Q₈	—	17	1	no	—
211	C₄×Q₈	—	17	1	no	—
212	C₄×Q₈	—	17	1	no	—
213	C₄×D₄	—	17	1	no	—
214	C₄×D₄	—	17	1	no	—
215	#117	—	34	1	no	—
216	#117	—	34	1	no	—
217	#107	—	17	1	no	—
218	C₄⋊C₈	—	17	1	no	—
219	C₄⋊C₈	—	17	1	no	—
220	C₄⋊C₈	—	17	1	no	—
221	#103	—	17	1	no	—
222	#103	—	17	1	no	—
223	#103	—	17	1	no	—
224	#103	—	17	1	no	—
225	#102	—	17	1	no	—
226	#102	—	17	1	no	—
227	#97	—	17	1	no	—
228	#97	C₂×D₁₇	1	1	yes	characteristic
229	C₄×C₈	—	17	1	no	—
230	C₁₇	#3402	1	1	yes	characteristic, complemented, Sylow
231	C₂×Q₈	—	17	1	no	—
232	C₂×Q₈	—	17	1	no	—
233	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
236	C₂×D₄	—	17	1	no	—
237	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
238	C₂²×C₄	—	34	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	4	1	no	—
242	Q₁₆	—	68	1	no	—
243	Q₁₆	—	68	1	no	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	4	1	no	—
246	SD₁₆	—	34	2	no	—
247	SD₁₆	—	68	1	no	—
248	D₈	—	68	1	no	—
249	C₂×C₈	—	17	1	no	—
250	C₂×C₈	—	17	1	no	—
251	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	—	34	1	no	—
264	C₄⋊C₄	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₄²	—	17	1	no	—
277	C₄²	—	17	1	no	—
278	C₄²	—	34	1	no	—
279	C₄²	—	34	1	no	—
280	C₄²	—	34	1	no	—
281	C₄²	—	34	1	no	—
282	C₄²	C₂²×D₁₇	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	34	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	34	1	no	—
292	Q₈	—	34	1	no	—
293	Q₈	—	68	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	34	1	no	—
297	D₄	—	34	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	17	1	no	—
302	C₂×C₄	—	17	1	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	17	1	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	68	2	no	—
309	C₂×C₄	—	68	2	no	—
310	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
311	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	34	1	no	—
315	C₈	—	34	1	no	—
316	C₈	#63254	1	2	yes	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	68	1	no	—
321	C₂²	—	68	1	no	—
322	C₂²	—	68	1	no	—
323	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	34	1	no	—
330	C₄	—	34	1	no	—
331	C₄	—	34	1	no	—
332	C₄	—	34	1	no	—
333	C₄	—	68	1	no	—
334	C₄	—	68	1	no	—
335	C₄	—	68	1	no	—
336	C₄	C₂×#63254	1	1	yes	characteristic
337	C₄	#10587691	1	1	yes	characteristic
338	C₂	—	8	1	no	—
339	C₂	—	136	1	no	—
340	C₂	#11763479	1	1	yes	characteristic
341	C₂	#11763608	1	1	yes	characteristic
342	C₂	#11763625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#497	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63228	C₂²
hypercenter	#497	D₁₇
hypocenter	C₁₇	#3402
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #11763533	C₂	semidirect
2. #11763528	C₂	semidirect
3. #11763202	C₂	semidirect
5. #11763033	C₂	semidirect
6. #11763029	C₂	semidirect
9. #11763017	C₂	semidirect
10. #11763016	C₂	semidirect
11. #11762937	C₂	semidirect
12. #11762911	C₂	semidirect
13. #11762910	C₂	semidirect
14. Q₁₆×Dic₁₇	C₂	semidirect
15. C₁₇×#497	C₂	semidirect
170. #497	D₁₇	semidirect
230. C₁₇	#3402	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11763533	2
#11763528	2
#11763202	2
#11763190	2
#11763033	2
#11763029	2
#11763026	2
#11763021	2
#11763017	2
#11763016	2
#11762937	2
#11762911	2
#11762910	2
Q₁₆×Dic₁₇	2
C₁₇×#497	2
#3402	17